Geometry & Topology

A very special EPW sextic and two IHS fourfolds

Maria Donten-Bury, Bert van Geemen, Grzegorz Kapustka, Michał Kapustka, and Jarosław Wiśniewski

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/gt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We show that the Hilbert scheme of two points on the Vinberg K3 surface has a two-to-one map onto a very symmetric EPW sextic Y in 5. The fourfold Y is singular along 60 planes, 20 of which form a complete family of incident planes. This solves a problem of Morin and O’Grady and establishes that 20 is the maximal cardinality of such a family of planes. Next, we show that this Hilbert scheme is birationally isomorphic to the Kummer-type IHS fourfold X0 constructed by Donten-Bury and Wiśniewski [On 81 symplectic resolutions of a 4–dimensional quotient by a group of order 32, preprint (2014)]. We find that X0 is also related to the Debarre–Varley abelian fourfold.

Article information

Source
Geom. Topol., Volume 21, Number 2 (2017), 1179-1230.

Dates
Received: 28 September 2015
Revised: 28 January 2016
Accepted: 3 March 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1510859177

Digital Object Identifier
doi:10.2140/gt.2017.21.1179

Mathematical Reviews number (MathSciNet)
MR3626600

Zentralblatt MATH identifier
1368.14016

Subjects
Primary: 14D06: Fibrations, degenerations 14J35: $4$-folds 14J70: Hypersurfaces 14K12: Subvarieties 14M07: Low codimension problems
Secondary: 14J50: Automorphisms of surfaces and higher-dimensional varieties 14J28: $K3$ surfaces and Enriques surfaces

Keywords
EPW sextics IHS fourfolds abelian varieties

Citation

Donten-Bury, Maria; van Geemen, Bert; Kapustka, Grzegorz; Kapustka, Michał; Wiśniewski, Jarosław. A very special EPW sextic and two IHS fourfolds. Geom. Topol. 21 (2017), no. 2, 1179--1230. doi:10.2140/gt.2017.21.1179. https://projecteuclid.org/euclid.gt/1510859177


Export citation

References

  • M F Atiyah, R Bott, A Lefschetz fixed point formula for elliptic complexes, II: Applications, Ann. of Math. 88 (1968) 451–491
  • A Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom. 18 (1983) 755–782
  • G Bellamy, T Schedler, A new linear quotient of ${\bf C}^4$ admitting a symplectic resolution, Math. Z. 273 (2013) 753–769
  • C Birkenhake, H Lange, Complex abelian varieties, 2nd edition, Grundl. Math. Wissen. 302, Springer (2004)
  • W Bosma, J Cannon, C Playoust, The Magma algebra system, I: The user language, J. Symbolic Comput. 24 (1997) 235–265
  • O Debarre, Annulation de thêtaconstantes sur les variétés abéliennes de dimension quatre, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987) 885–888
  • L Dixon, J A Harvey, C Vafa, E Witten, Strings on orbifolds, Nuclear Phys. B 261 (1985) 678–686
  • I V Dolgachev, Classical algebraic geometry: a modern view, Cambridge University Press (2012)
  • I Dolgachev, Corrado Segre and nodal cubic threefolds, preprint (2015)
  • I V Dolgachev, D Markushevich, Mutually intersecting planes in ${\mathbb P}^5$, Enriques surfaces, cubic $4$–folds, and EPW-septics, Oberwolfach Rep. 7 (2010) 1603–1605
  • M Donten-Bury, J A Wiśniewski, On $81$ symplectic resolutions of a $4$–dimensional quotient by a group of order $32$ (2014) To appear in Kyoto J. Math.
  • D Eisenbud, S Popescu, C Walter, Lagrangian subbundles and codimension $3$ subcanonical subschemes, Duke Math. J. 107 (2001) 427–467
  • B van Geemen, J Top, An isogeny of $\mathrm{K3}$ surfaces, Bull. London Math. Soc. 38 (2006) 209–223
  • D R Grayson, M E Stillman, Macaulay2, a software system for research in algebraic geometry Available at \setbox0\makeatletter\@url http://www.math.uiuc.edu/Macaulay2/ {\unhbox0
  • P Griffiths, J Harris, Principles of algebraic geometry, Wiley-Interscience, New York (1978)
  • G Hoehn, G Mason, Finite groups of symplectic automorphisms of hyperkähler manifolds of type $K3^{[2]}$, preprint (2014)
  • G Kapustka, On IHS fourfolds with $b_2=23$, Michigan Math. J. 65 (2016) 3–33
  • J Kollár, Shafarevich maps and plurigenera of algebraic varieties, Invent. Math. 113 (1993) 177–215
  • H Maschke, Ueber die quaternäre, endliche, lineare Substitutionsgruppe der Borchardt'schen Moduln, Math. Ann. 30 (1887) 496–515
  • U Morin, Sui sistemi di piani a due a due incidenti, Atti Istituto Veneto 89 (1930) 907–926
  • K G O'Grady, Involutions and linear systems on holomorphic symplectic manifolds, Geom. Funct. Anal. 15 (2005) 1223–1274
  • K G O'Grady, Irreducible symplectic $4$–folds and Eisenbud–Popescu–Walter sextics, Duke Math. J. 134 (2006) 99–137
  • K O'Grady, Double covers of EPW-sextics, Michigan Math. J. 62 (2013) 143–184
  • K G O'Grady, Pairwise incident planes and hyperkähler four-folds, from “A celebration of algebraic geometry” (B Hassett, J McKernan, J Starr, R Vakil, editors), Clay Math. Proc. 18, Amer. Math. Soc., Providence, RI (2013) 553–566
  • Y G Prokhorov, Fields of invariants of finite linear groups, from “Cohomological and geometric approaches to rationality problems” (F Bogomolov, Y Tschinkel, editors), Progr. Math. 282, Birkhäuser, Boston (2010) 245–273
  • R Varley, Weddle's surfaces, Humbert's curves, and a certain $4$–dimensional abelian variety, Amer. J. Math. 108 (1986) 931–951
  • E B Vinberg, The two most algebraic $\mathrm{K3}$ surfaces, Math. Ann. 265 (1983) 1–21